On smoothness, tangent cones, and the metric geometry of definable sets
André Gadelha Rocha, José Edson Sampaio
TL;DR
The paper develops a comprehensive framework to characterize $C^1$ smoothness of definable and real analytic sets in o-minimal structures via tangent cones ($C_3$, $C_4$, $C_5$) and metric regularity notions such as $LNE$. It proves a suite of equivalences: $C^1$ smoothness is equivalent to combinations of Lipschitz regularity, linearity of tangent-cones, and their continuous dependence on the base point, with sharper results in the real analytic setting and connections to the Nash mapping. It also provides explicit counterexamples that delineate sharp boundaries, including parametrised analytic images and zeros of harmonic functions, and extends the dialogue to the Nash transformation and multiplicity theory. Overall, the work gives a unified, metric-geometric view of smoothness for definable and real analytic sets, with practical implications for regularity criteria and geometric analysis in o-minimal contexts.
Abstract
In this paper, we present several definitive characterizations of the $C^1$ smoothness of definable sets in terms of their tangent cones and some other metric properties. In particular, we recover some of the beautiful characterizations presented by Ghomi and Howard (2014) and by Kurdyka, Le Gal, and Nhan (2018). For instance, we prove that for any $X\subset \mathbb{R}^n$ that is a locally closed $d$-dimensional definable set in an o-minimal structure, the following items are equivalent: (1) $X$ is Lipschitz normally embedded (LNE), $C_3(X,p)$ is a $d$-dimensional linear subspace for any $p\in X$ and depends continuously on $p$; (2) For each $p\in X$, $X$ is Lipschitz regular at $p$ and $C_4(X,p)$ is a $d$-dimensional linear subspace; (3) $X$ is a topological manifold and for each $p\in X$, $X$ is LNE at $p$ and $C_4(X,p)=C_3(X,p)$; (4) $X$ is a topological manifold, and $C_5(X,p)$ is a $d-$dimensional subset for any $p\in X$; (5) $X$ is $C^1$ smooth.